My favorite trail in the park is the Skyhy Ridge Walk. The loop I hike more than any other is BMRR South Trail to the TCI Connector to the Skyhy Ridge to the Smythe that connects back to the BMRR South. That loop is a little under 3 miles but its has some elevation change that make it challenging and fun. When I have time I will extend this loop and go to Haskell Treehouse before I head back down the Smythe. The Smythe trail is steeper than the Songo so it saves some milage but the Songo is an easier trail to get up or down the ridge. The Skyhy Ridge makes some people little nervous because it is so narrow.
Whatever adventure you pick in Red Mountain Park make sure you grab a map before you go! They have a map online that you can print out. Trust me it will come in handy otherwise you might get lost and end up hiking a lot further than you had planned. Also check out their website for special events. They offer guided group hikes and HOGA (Hiking & Yoga) you just have to sign up in advance because they limit the size of the groups. I am dying to check out the HOGA. They just started offering it and I signed us up for the one last weekend but it was canceled due to weather. Maybe next month!